Constrained School Choice



Proof By Theorem 1 of Kesten (2006), τ = γ. Hence, Oτ (P, k) = Oγ(P, k). By
Lemma 1 of Kesten (2006), f is Ergin-acyclic. So, from Theorem 6.5, S(P) =
Oγ (P, k) =
Oτ (P,k).                                                                           ■

Proof of Theorem 6.8 Follows from Lemmas B.10 and B.11.

Lemma B.12 Let the priority structure f admit an X-cycle. Let 1 k m. Then,
there is a school choice problem
P with a Pareto inefficient equilibrium outcome in the
game
Γτ(P, k), i.e., for some Q Eτ(P, k), τ(Q) PE(P).

Proof Since f admits an X -cycle, we may assume, without loss of generality, that
(a) f
s1(ij) < fs1(i1) < fs1(i2) for each j I1 := {3, . . . , qs1 + 1} and

(b) fs2(ij) < fs2(i2) < fs2(i1) for each j I2 := {qs1 + 2, . . . ,qs1 + qs2}.

Consider students’ preferences P defined by Pi1 := s2, s1, Pi2 := s1, s2, Pij := s1 for
j I1, Pij := S2 for j I2, and Pij := 0 for all j 1 + qs2 + 1,. .., n}.

Consider Q Q(k)I defined by Qi1 := s1, Qi2 := s2, and Qi := Pi for all i I \{i1 , i2}.
One easily verifies that at τ(Q) all students in
{i3, i4, . . . , iqs1 +qs2 } are assigned to their
favorite school. Also, τ(Q)(i
1) = s1 and τ (Q)(i2) = s2. It is obvious that at τ(Q)
students i
1 and i2 would like to swap their seats, i.e., τ (Q) PE(P). Nevertheless, there
is no unilateral deviation for either of the two students to obtain the other seat. Hence,
Q
Eτ(P,k).                                                              ■

Proposition B.13 Let 1 k m. If for some school choice problem P there exists
Q Eτ (P, k) such that τ (Q) / PE(P) then f admits an X -cycle.

Proof Let Q Eτ(P, k) be such that τ(Q) / PE(P). In view of Proposition B.9 we may
assume without loss of generality that k = 1 and for each student i
I, Qi = τ (Q)(i).
For any school s
S and any profile Q QI, let As(Q) be the set of students to which
school s points whenever school s is part of a cycle,
i.e.,

As(Q) := {i I : there is a step l of TTC(Q) with i = e(Q, l, s) and s F(Q, l, i)} .

Step 1 There exist p 2, a set of students CI = {i1 , . . . , ip}, and a set of schools
CS = {s1, . . . , sp} such that

(a) for each student ir CI, srPir sr+1 = τ (Q)(ir) (where ip+1 = i0),

(b) for each school s CS, As(Q) = τ(Q)(s) = qs, and

(c) for any two distinct schools s, s' Cs, As(Q) As'(Q) = 0.

40



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